G. Polya and G. Szego. Problems and Theorems in Analysis II. Springer-Verlag, 1976. Home/Search Document Not in Database Summary Related Articles Check |
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Strictly Positive Definite Kernels On The Hilbert Sphere - Menegatto (1994) (6 citations) (Correct)
....SEMI METRIC SPACES Let A = A ij ) be a matrix with real entries. If B = B ij ) is a matrix of the same size, then the Hadamard product or Schur product A ffl B is defined by entrywise multiplication, i.e. A ffl B) ij : A ij B ij . A well known result of Schur (p. 100 in P olya and Szego 11 ) states that A ffl B is a nonnegative definite matrix whenever A and B are nonnegative definite. As a consequence of this result, it can be proved (p.101 in Horn and Johnson 7 ) that the matrix (exp A ij ) is nonnegative definite whenever A is nonnegative definite. As a matter of fact, exp ....
....2k 1 t = 1 X k=1 (2k) 2 2k (k ) 2 (2k 1) 1 Gamma cos 2k 1 t) Consequentely, d : S 1 Theta S 1 Gamma [0; is a conditionally negative definite kernel. It is not strictly conditionally negative definite as can be seen by taking four points STRICTLY POSITIVE DEFINITE KERNELS 11 x 1 ; x 2 ; x 3 and x 4 in S 1 with d 1 (x i ; x j ) 2 for 1 i j 4. The above remarks together with Theorem 3.6 imply our final result: Theorem 3.7. The following hold: a) If F 2 DM then F ffi d1 is a strictly conditionally negative definite kernel. b) If F 2 CM then F ffi d1 is a ....
G. P'olya and G. Szego, Problems and Theorems in Analysis II, Springer-Verlag, New York, 1976.
Interpolation On The Complex Hilbert Sphere Using Positive.. - Menegatto (1997) (Correct)
....exp(A ) The main results in this paper depend on the invertibility of Schur exponentials. It is well known that exp(A) is nonnegative definite if either A is nonnegative definite or almost nonnegative definite in the sense that it is hermitian and P n ; 1 c c A 0 when P n =1 c = 0 ( 6] [13]) Thus, to guarantee invertibility of exp(A) in these two cases, we only need to know under what circumstances the Schur exponentials will be positive definite. In Lemmas 2.1 and 2.2 below, we present two conditions under which Schur exponentials are positive definite. Before proving these ....
P'olya, G. and G. Szego, Problems and Theorems in Analysis II, Springer-Verlag, New York, 1976.
Short Interval Results For ...-Free Values Of Irreducible.. - Filaseta (Correct)
.... (and slightly improved) a theorem of Halberstam and Roth [5] which stated that for every ffl 0 and for h = x 1= 2k) ffl , there is a k Gammafree number in the interval (x; x h] We note that if k n 1 and the greatest common divisor of the coefficients of f(z) is 1, then (1) holds (cf. [11]) An improvement on (2) follows from the work of Huxley and Nair [7 (take t = k Gamma g 1 in Theorem A) Their work implies that if k n 1 3, then one can take (3) h = cx n= 2k Gamman 2) One can further reduce h by a power of log x. A direct application of the techniques in [15] do ....
G. P'olya and G. Szego, Problems and Theorems in Analysis II (Part VIII, prob. 86, p. 129), Springer-Verlag, Berlin, 1976.
Solution Of One Problem Of G. Pólya - Jankovic (1996) (Correct)
....2) In the case r = ae, determine the directions in which he she should look in order to see out of the forest. We assume that the ray tangent to a tree is not blocked by that tree. The above problem was studied for the first time by G. P olya in [1] where he proved that lim R 1 aeR = 1: In [2] (problem 239) the following estimate 1 p R 2 1 6 ae 1 R in the case when R is integer is obtained by method of A. Speiser. In some more recent books (see [3] 4] the same estimate is obtained using Minkowski theorem concerned with the existence of the lattice point in the set which ....
G. P'olya, G. Szego, Problems and Theorems in Analysis II, Springer-Verlag, 1976.
Periodicity and Repetition in Combinatorics on Words - Wang (2004) (Correct)
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G. Polya and G. Szego. Problems and Theorems in Analysis II. Springer-Verlag, 1976.
Continued Fractions and Linear Recurrences - Lenstra, Jr., Shallit (2001) (Correct)
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G. Polya and G. Szego. Problems and Theorems in Analysis II. Springer-Verlag, 1976.
The Algebra and Its Connection With the Quaternion Algebra - Razon (Correct)
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G. Polya and G. Szego, Problems and Theorems in Analysis II, Springer-Verlag, New York, 1976.
Squarefree Values Of Polynomials - Filaseta (Correct)
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G. P'olya and G. Szego, Problems and Theorems in Analysis II, Springer-Verlag, New York, 1976.
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